The similarity is a coincidence. The words "Roma" and "Romani" are ultimately derived from the word "Rom", which means "man" in the Romani language itself. More specifically, "Romani" is the feminine form of "Romano", which means "of or pertaining to the Roma people."

The term "gypsy" comes from the (mistaken) belief that the Romani people originated in Egypt. Of course, they actually have their roots in India. You might have noticed that the same misconception (probably) gives rise to cognates in other languages, such as "gitano" in Spanish (I say probably because there is speculation that it could actually come from "Cingane", a term for a Christian group sometimes associated with the Romani).

edit: Spelling.

On a related note, why was Rome called Rome? Is it really just named after Romulus, or was there some other etymological root?

If I could piggy back a question, What is a good survey book on the history of the Romani?